ACFT DA90 TAXIED ACROSS ACTIVE RWY WITHOUT CLRNC.

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DIVERTED TO EWR DUE TO DELAYS INTO TEB; OUR DEST. AFTER LNDG ON RWY 22L; OUR TAXI INSTRUCTIONS WERE TO TAXI N BTWN THE 2 RWYS ON TXWY P AND HOLD SHORT OF TXWY W; WHICH CROSSES RWY 22R AT THE APCH END. AFTER RECEIPT OF THIS CLRNC; THE FO WENT OFF FREQ TO TALK TO THE FBO ON THE OTHER RADIO. WHILE THE FO (WHO HAD THE TAXI DIAGRAM) WAS OFF FREQ; I RECEIVED A REVISED CLRNC; WHICH IN PART STATED 'L TURN ON TXWY P; CROSS RWY 29' AND I THOUGHT A CLRNC TO CROSS RWY 22R. I READ BACK; 'ROGER; L ON TXWY P AND CLR TO CROSS THE PARALLEL.' AT THIS TIME I WAS APCHING TXWY Y; WHICH HAD BOTH TXWY P AND TXWY Y SIGNAGE TO IDENT THE INTXN. BEING UNFAMILIAR WITH THE ARPT; HAVING FORGOTTEN I WAS ALREADY ON TXWY P AND NOT HAVING MY FO IN THE LOOP; I MISTOOK TXWY Y FOR TXWY P; MADE A L TURN AND CROSSED RWY 22R; WHICH WAS ACTIVE. ONCE ON THE OTHER SIDE; THE CTLR INFORMED ME THAT WAS NOT RWY 29; WHICH I ALREADY KNEW BUT THOUGHT I WOULD EVENTUALLY HAVE TO CROSS IT TO GET TO PARKING. MY MISTAKE WAS ALLOWING MYSELF TO GET RUSHED INTO SOMETHING BEFORE I HAD A CHANCE TO CONFIRM THE VALIDITY OF WHAT I WAS DOING WITH THE TAXI DIAGRAM. A SUBSEQUENT REVIEW OF THE TAXI DIAGRAM SHOWED TXWY P EVENTUALLY MAKING A DOGLEG L TURN (HENCE THE CTLR'S L TURN INSTRUCTIONS) BEHIND AND CLR OF RWY 22R AND XING RWY 29. I BELIEVE A CONTRIBUTING FACTOR WAS THE CTLR'S INSTRUCTIONS TO 'TURN L ON TXWY P' WHEN 'CONTINUE ON TXWY P' WOULD HAVE BEEN MORE APPROPRIATE.